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As Anthropic prepares to IPO for over $1 trillion, a reminder of Dario Amodei’s words shortly after founding the company:
“Distributed human labor is being used to train AI models by large, centralized actors, who concentrate the resulting profits while in the long run making the human labor obsolete.”
In other words, this IPO will be built off the backs of the world’s creatives - who will get nothing out of it except obsolescence.
It is a totally unjust transfer of wealth and opportunity from artists, writers, musicians & other creatives to a tiny tech elite.
I hope it is remembered as such.
(this quote is from a document he wrote that was unsealed in Bartz v. Anthropic)
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In case anyone missed this, OpenAI has proved the Kakeya conjecture in all dimensions.
Hong Wang got the Fields medal in 2026 for proving the Kakeya conjecture in dimension 3.
In fact OpenAI appears to have committed, in mathematics terms, what may be fairly termed a massacre, proving a far stronger result. It has proved the local smoothing conjecture, which sits at the very top of the tower of conjectures as described in the Quanta article in the reply.
I am not an expert in this area -- so happy to be corrected -- but the methods appear to be PDE based and basically bypasses a lot of work of the last two decades. This is the equivalent of Arbreans showing up to earth with an everything killer and explaining to us we have been doing warfare all wrong.
Announcing three major scientific discoveries, one each in mathematics (harmonic analysis), computer science (complexity theory), and physics (quantum field theory).
AI starting to help us improve our understanding of the universe! openai.com/index/building-sc…
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There is something odd about this. With all due respect to past achievement, setting up a collaboration where the lead is 85 and one member is 88 doesn't look optimised for science. Why not get the best modern QFT experts here? I could name the people who should be on this.
The physics collaboration will understand and explain a result that gives a mathematically rigorous existence proof of certain interacting quantum field theories containing gauge fields and fermions in four dimensions, including Quantum Chromodynamics (QCD) and a unified completion of the Standard Model of particle physics.
As a subproblem, this work also includes a proof of the existence of pure Yang–Mills theory, which is part of the Quantum Yang–Mills Theory Clay Mathematics Institute Millennium Prize problem. These proofs are all formally verified in Lean.
Our models produced a natural language proof for Yang–Mills existence, and then subsequently produced a proof for QCD After the formalization completed, we reached out to David Gross. David will lead a group including Michael Douglas, Arthur Jaffe, Igor Klebanov, and Nikita Nekrasov that will shepherd this work, and we will defer to them on the release of materials.
Gross’s work on asymptotic freedom in Yang-Mills theories helped establish QCD as the theory of the strong interactions and earned him a share of the 2004 Nobel Prize in Physics. Jaffe formulated the Quantum Yang–Mills Theory Millenium Prize problem with Witten and also envisioned the idea of Millennium Prize problems and co-founded the Clay Mathematics Institute. Douglas has already been a steward of this problem over the last 20 years; Klebanov helped pioneer the application of gauge–string duality to confinement, an important property of QCD, and now directs the Simons Collaboration on Confinement and QCD Strings; and Nekrasov has done groundbreaking work on exact methods for generalizations of Yang–Mills theory.
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An internal model at OpenAI has produced a claimed refutation of ETH! Russell Impagliazzo, Mohan Paturi, and I have been asked to help organize an open and independent examination of this claim. 1/4
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Wooow – Kakeya conjecture solved in all dimensions by OpenAI in a 50 page paper!
Actually this paper proves a much more general conjecture. Looks from afar like very significant progress on that topic!
github.com/openai/math/blob/…
openai.com/index/building-sc…
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Our letter to the community invites ideas for organizing this effort and, separately, for addressing the broader questions raised by AI-generated results:
docs.google.com/document/d/1…
OpenAI's announcement is here: openai.com/index/building-sc…
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As was well-known, and circulated a number of times even on here, Ethan Sussman has been writing up existence of YM4 publicly via Balaban’s approach, and has posted existence of QED3 and YM3 and an old preliminary outline of YM4 on his repo, working with countless community members in understanding the arguments. I kept telling him to not be too deferential and modest because he would be publicly scooped at any opportunity. Instead of contacting him to work with him, as has been suggested publicly many times, the people at OpenAI chose to do this.
“Our work builds on Balaban’s approach in constructive quantum field theory. As a subproblem, this work also includes a proof of the existence of pure Yang–Mills theory, which is part of the Quantum Yang Mills Theory Clay Mathematics Institute Millennium Prize problem.”
We are announcing three new discoveries in mathematics, computer science, and physics about harmonic analysis, complexity theory, and quantum field theory.
openai.com/index/building-sc…
MC retweeted
they should IMMEDIATELY release this
we do no need shepherds
The physics collaboration will understand and explain a result that gives a mathematically rigorous existence proof of certain interacting quantum field theories containing gauge fields and fermions in four dimensions, including Quantum Chromodynamics (QCD) and a unified completion of the Standard Model of particle physics.
As a subproblem, this work also includes a proof of the existence of pure Yang–Mills theory, which is part of the Quantum Yang–Mills Theory Clay Mathematics Institute Millennium Prize problem. These proofs are all formally verified in Lean.
Our models produced a natural language proof for Yang–Mills existence, and then subsequently produced a proof for QCD After the formalization completed, we reached out to David Gross. David will lead a group including Michael Douglas, Arthur Jaffe, Igor Klebanov, and Nikita Nekrasov that will shepherd this work, and we will defer to them on the release of materials.
Gross’s work on asymptotic freedom in Yang-Mills theories helped establish QCD as the theory of the strong interactions and earned him a share of the 2004 Nobel Prize in Physics. Jaffe formulated the Quantum Yang–Mills Theory Millenium Prize problem with Witten and also envisioned the idea of Millennium Prize problems and co-founded the Clay Mathematics Institute. Douglas has already been a steward of this problem over the last 20 years; Klebanov helped pioneer the application of gauge–string duality to confinement, an important property of QCD, and now directs the Simons Collaboration on Confinement and QCD Strings; and Nekrasov has done groundbreaking work on exact methods for generalizations of Yang–Mills theory.
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Two physicists asked AI to choose a problem on its own, solve it and write a paper that could be published in a prestigious journal. In two hours, AI produced a paper with a novel, significant result.
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Since I'm now blocked by bluey from replying to this thread, I answer here by quote tweet
the openAI construction has more to do with convex integration than anything else.
In convex integration, the oscillations are typically built by hand and the iteration procedure creates wild weak solutions. The non-uniqueness construction of Albritton–Brué–Colombo used an unstable vortex to split solutions apart. The Córdoba–Martínez-Zoroa singularities amplify vorticity across scales by strain. In the openAI paper, a fluid instability is clever made so that the flow itself generates the oscillations: they grow by drawing energy from the background shear. The Reynolds stress then becomes the force that creates the collapsing smooth vortex. Using a physical instability to create the stress in this way has no precedent, and it is what produces a smooth force rather than a merely rough one.
Some on these ideas are in the literature (e.g. Leibovich-Stewartson 1983) but the way they are combined is definitely insane and definitely nowhere in the literature previously.
I also don't know of anyone suggesting that convex integration would play a role in the creation of a blow-up solution like this.
On a technical level, the paper is unbelievable, maybe beyond what a human could have come up with but not beyond human understanding: there are several different groups of pde experts working on understanding it who say that they more or less understand now. I expect digestions soon
td;lr: anyone saying that this OAI result is unsurprising is bullshitting
sorry!
Or it’s reasonable. Especially considering what the models were trained on, and that they spent $15M in compute instead and of $15M in consulting fees for specialist mathematicians to just do research (no teaching no service).
I don’t know why mathematicians want to pretend that no relevant progress was made before OAI released their solution.
MC retweeted
OpenAI's proof of the Haldane gap in the S=1 antiferromagnetic Heisenberg chain: Expository notes
haltasaki.github.io/writings…
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My friend and colleague Bill Banks has just uploaded a very nice exposition on the proof of no Siegel zeros:
arxiv.org/abs/2610.15483
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“Our experiments suggest that small AI models may be months away from being capable enough to self-replicate on commercial IoT devices, like smart ovens and off-brand Android TV boxes. A replicator like this would be challenging to shut down, since there’s no centralized point to shut off - though it may be slow (with a doubling time of days). Unlike non-AI malware, it could also figure out unfamiliar devices on its own. In our evals, a 9B model got inference running on 8 of the 9 hardware types we tested.
Models of 30B parameters and under can already complete many of the individual steps for replication today, though they make mistakes when composing them end to end. In our simplified evals, GLM-5.2 and stronger models are capable of end to end replication. If small models improve at their historical rate, they’ll match this capability level by the end of this year. Small weight-based replicators may be possible soon after.” -@bucketofkets @willzhou97
anthropic.com/research/self-…
MC retweeted
Amodei's essay is really called "Machines of Loving Grace" and he thinks AI is going to cure almost all disease, double the human life span, democratize world politics, and possibly raise GDP of sub Saharan African countries to the level of China in "5-10 years."
Dario Amodei considers people opting out of AI-driven transformation of human life as a problem, perhaps a dystopian problem, needing to be "solved", on par with anti-vax. This is from the same essay, 'Machines of Loving Grace', in which he advocates that we become pets of the machines, and suggests video games as one of several alternate vectors of meaning to replace meaningful struggle and challenge.
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Important piece from Microsoft AI CEO @mustafasuleyman. Training models to think of themselves as human (as Anthropic does Claude) with concepts of an inner self, psychological wellbeing, rights and presumably grievances, is a potential existential risk.
project-syndicate.org/onpoin…
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Mathematicians need to be Clearpilled. Writing essays will not solve your problems. Crying in the casino won't solve your problems. Identify your enemy and recognize they are enemy of human society itself. A narrow focus on trying to save mathematics solves net total of nothing.
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Update from Scott Aaronson: Some AI companies are now using their latest internal models to take a run at breaking 'important cryptographic protocols and primitives.'
He also adds this, to which I agree:
'Also noteworthy is the striking under-representation of cryptographic breakthroughs among the 722 mathematical results OpenAI published. I've witnessed first-hand the US government censoring academic quantum cryptanalysis results. Backroom interventionism is my base case.'
Things are probably further along than it seems. This is true in many areas right now. There are also persistent rumors that the OpenAI math release yesterday was only the first batch of three.
